Tangential Schur, Pick matrices, and J-inner factors

This page describes the experimental tangential-Schur layer in lattice-dsp. It is deliberately finite-dimensional and definite. The goal is to give the matrix/MIMO lattice examples a transparent interpolation-theory baseline without claiming a complete generalized indefinite Schur solver or the full manifold parametrization algorithms used in the Hanzon–Olivi–Peeters/Marmorat line of work.

Schur functions and the RKHS/Pick viewpoint

A matrix-valued Schur function is an analytic transfer matrix \(S(z)\in\mathbb{C}^{p\times q}\) on the unit disk with

\[\lVert S\rVert_\infty \le 1.\]

Equivalently, the de Branges–Rovnyak kernel

\[K_S(z,w) = \frac{I_q-S(z)^H S(w)}{1-\overline z w}\]

is positive. Right tangential interpolation prescribes only selected input directions:

\[S(z_i)U_i = V_i, \qquad |z_i|<1.\]

Here \(U_i\in\mathbb{C}^{q\times r_i}\) contains input directions and \(V_i\in\mathbb{C}^{p\times r_i}\) contains the desired output vectors. A rank-one condition uses a single vector \(u_i\) and value \(v_i\).

The finite Pick matrix implemented by the package is exactly the Gram matrix of this kernel tested on the tangential columns:

\[P_{ij} = U_i^H K_S(z_i,z_j) U_j = \frac{U_i^H U_j - V_i^H V_j} {1-\overline{z_i}z_j}.\]

The definite Schur-class feasibility condition is

\[P\succeq 0.\]

This is the RKHS reason that the Pick matrix appears: feasible interpolation data must make the finite kernel Gram matrix positive semidefinite. In the implemented module this is used as a finite certificate and conditioning diagnostic, not as a black-box synthesis of all rational solutions.

Constant Schur solution helper

The package includes constant_schur_solution for the special case where the data are compatible with a constant matrix contraction. It solves

\[S_0 [U_1\;\cdots\;U_n] = [V_1\;\cdots\;V_n]\]

by a pseudoinverse and then checks both interpolation residual and contractivity. This is not a general Schur-synthesis routine. It is a useful sanity check for examples because the exact solution is known.

Lossless/all-pass functions and J-inner matrices

For a square transfer matrix \(Q(z)\in\mathbb{C}^{p\times p}\), a stable lossless or inner all-pass function satisfies

\[Q(e^{j\omega})^H Q(e^{j\omega}) = I_p\]

for almost every point on the unit circle. In state-space/DSP language this is an energy-preserving stable all-pass system. The package has a causal runtime for such forward matrix-lattice all-pass filters through OnlineMatrixLatticeAllPass.

Tangential Schur theory often works with a larger signature matrix

\[\begin{split}J= \begin{bmatrix} I_p & 0 \\ 0 & -I_q \end{bmatrix}\end{split}\]

and with \(J\)-inner transfer matrices \(\Theta(z)\) satisfying

\[\Theta(e^{j\omega})^H J \Theta(e^{j\omega}) = J.\]

These \(J\)-inner matrices are the natural objects in linear-fractional Schur transformations. They are related to, but not identical with, the square all-pass runtime filters in the package: the runtime filters act on physical signals, while the \(J\)-inner factors act on graph vectors \([v;u]\) and parameterize interpolation transformations.

Potapov–Blaschke factors

A scalar Blaschke factor for a point \(a\) in the unit disk is

\[b_a(z)=\frac{z-a}{1-\overline a z}, \qquad |b_a(e^{j\omega})|=1.\]

A Potapov–Blaschke factor is the matrix/signature analogue. For strict rank-one data \(S(a)u=v\), define the graph vector

\[\begin{split}\xi=\begin{bmatrix}v\\u\end{bmatrix}, \qquad \xi^H J\xi=\lVert v\rVert_2^2-\lVert u\rVert_2^2<0.\end{split}\]

The package forms the \(J\)-orthogonal projection

\[P_\xi = \xi(\xi^H J\xi)^{-1}\xi^H J\]

and the elementary factor

\[\Theta_a(z)=I+(b_a(z)-1)P_\xi.\]

The two identities checked in code are

\[\Theta_a(e^{j\omega})^H J\Theta_a(e^{j\omega}) = J, \qquad \Theta_a(a)\xi=0.\]

Products of such factors remain \(J\)-inner on the unit circle. In the package, TangentialPotapovFactor and PotapovProduct are used as inspectable diagnostics: they expose the same algebraic invariants that underlie matrix lossless/lattice theory, without claiming full generalized Schur synthesis.

Schur parameters

In the scalar lattice/AR part of the package, Schur parameters are the familiar reflection/PARCOR coefficients. Bounded scalar Schur parameters \(|k_i|<1\) give a stable all-pole lattice denominator.

In multivariable tangential-Schur theory, a chart is usually described by chosen interpolation points and directions together with contractive interpolation values. Informally, the values \(v_i\) in

\[Q_i(w_i)u_i = v_i, \qquad \lVert v_i\rVert < 1,\]

play the role of Schur parameters in that chart. The Hanzon–Olivi–Peeters and Marmorat literature uses such parameters to build overlapping local parametrizations of fixed-degree stable all-pass/lossless systems and to connect them with balanced state-space realizations.

The current package does not implement that full recursive chart algorithm. It implements the finite Pick matrix, constant-solution checks, and elementary rank-one \(J\)-inner/Potapov factors that make the surrounding algebra visible and testable.

Relation to Hanzon–Olivi–Peeters/Marmorat algorithms

The implemented utilities are in the same mathematical ecosystem as the real/complex tangential Schur algorithms used for manifolds of multivariable stable all-pass systems, but they are not the same algorithm.

What those works use, in broad terms, is a recursive tangential-Schur procedure: chosen chart data define interpolation conditions; each step applies a linear-fractional transformation associated with a degree-one \(J\)-inner factor; the resulting Schur parameters provide local coordinates on a manifold of fixed-degree lossless/all-pass systems, often with associated balanced state-space realizations and chart-selection strategies.

What lattice-dsp implements in this release is narrower:

  • finite right-tangential data containers;

  • the definite Pick/RKHS Gram matrix \(P\succeq0\);

  • residual checks for candidate transfer matrices;

  • constant contractive solutions when the data are compatible with one;

  • elementary rank-one Potapov–Blaschke/\(J\)-inner factors;

  • dense-grid \(J\)-inner residual diagnostics.

Not implemented here:

  • recursive extraction of Schur parameters from an arbitrary matrix inner function;

  • an atlas of overlapping charts for fixed McMillan-degree lossless systems;

  • balanced canonical state-space forms generated by the tangential-Schur recursion;

  • manifold optimization for matrix \(H_2\) approximation;

  • generalized indefinite or singular tangential interpolation.

So the precise statement is: this module provides a tested finite definite Pick/Potapov baseline that is compatible with the language used in that literature, but it should not be described as the full Marmorat–Hanzon–Olivi– Peeters tangential-Schur manifold algorithm.

Verification philosophy

The tangential-Schur implementation is tested through identities that are known mathematically, rather than through one visually pleasing example. The core verification cases are:

  • scalar data reduce to the classical Nevanlinna–Pick matrix;

  • one-node rank-one data reduce to the norm condition \(\lVert v\rVert_2\le\lVert u\rVert_2\);

  • data generated by a known constant contraction \(S_0\) have \(P\succeq0\) and recover \(S_0\) when the tangential directions span the input space;

  • diagonal MIMO data with coordinate directions decompose into independent scalar Pick blocks;

  • feasible nonconstant scalar Schur data, for example \(S(z)=c b_a(z)\), pass the Pick test but are rejected by the constant-solution helper;

  • tangential-column changes \(U_i\mapsto U_iQ_i\), \(V_i\mapsto V_iQ_i\) transform the Pick matrix by a block-unitary congruence and preserve its eigenvalues;

  • Potapov projections satisfy \(P_\xi^2=P_\xi\) and \(P_\xi^HJ=JP_\xi\);

  • elementary Potapov factors and their products satisfy the boundary identity \(\Theta(e^{j\omega})^HJ\Theta(e^{j\omega})=J\) on dense frequency grids.

These tests are documented in Tangential Schur verification map. They are meant to make the finite definite baseline credible while still keeping clear that full generalized Schur synthesis is outside the release scope.

Causality and data use

Tangential Schur/Pick routines operate on finite interpolation data. They are not online time-series predictors and do not consume samples one at a time. They are best understood as offline certificates and construction diagnostics.

This is different from MIMOLatticePredictor or OnlineMatrixLatticeAllPass: those objects are causal runtimes. Tangential Schur utilities explain part of the mathematical background for matrix-lattice/all-pass structures, while the online classes implement sample-by-sample DSP behavior.

Scope boundary

Implemented in this release:

  • right tangential data containers;

  • definite tangential Pick matrices and PSD checks;

  • residual checks for candidate transfer matrices;

  • constant contractive solutions when the data are compatible with one;

  • elementary rank-one Potapov/J-inner factors;

  • J-inner residual checks for individual factors and products.

Not claimed in this release:

  • full generalized indefinite Schur interpolation;

  • automatic synthesis of all rational Schur solutions from arbitrary Pick data;

  • boundary interpolation and singular cases;

  • complete MIMO inner–outer factorization.

The module is therefore a careful baseline rather than a black-box matrix-valued interpolation solver.